be a non-zero cuspidal Hecke eigenform of half-integral weight \(k+1/2\)  and level \(N\)  with Dirichlet character \(\varepsilon \pmod {N}\) , and let \(t\)  be a square-free integer such that \(a(t)\ne 0\) . The Shimura correspondence [1]} lifts \(f\)  to a Hecke eigenform \(\mathrm {Sh}_t(f)\)  of weight \(2k\)  for the group \(\Gamma _0(N/2)\)  with character \(\varepsilon ^2\) . Let us write
\(\mathrm {Sh}_t(f)=\sum _{n\ge 1} A_t(n)q^n,\) 
